A complete resolution of \(M\in \operatorname{mod}(R)\)  is a diagram \(T\overset{\tau }{\rightarrow }P\overset{\simeq }{\rightarrow } M\) , where \(P\overset{\simeq }{\rightarrow }M\)  is a projective resolution, \(T\)  is a totally acyclic complex of free \(R\) -modules, and \(\tau _i\)  is an isomorphism for all \(i\gg 0\) . It is known that \(M\)  has finite Gorenstein dimension if and only if it has a complete resolution [1]}.
